Solve 1/n+2+3/3n+1=2/n+1

$$\frac { 1 } { n + 2 } + \frac { 3 } { 3 n + 1 } = \frac { 2 } { n + 1 }$$

Solve for n

$$n=3$$

Steps for Solving Linear Equation

Variable n cannot be equal to any of the values -2,-1,-\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by \left(n+1\right)\left(n+2\right)\left(3n+1\right), the least common multiple of n+2,3n+1,n+1.

$$\left(n+1\right)\left(3n+1\right)+\left(n+1\right)\left(n+2\right)\times 3=\left(n+2\right)\left(3n+1\right)\times 2$$

Use the distributive property to multiply n+1 by 3n+1 and combine like terms.

$$3n^{2}+4n+1+\left(n+1\right)\left(n+2\right)\times 3=\left(n+2\right)\left(3n+1\right)\times 2$$

Use the distributive property to multiply n+1 by n+2 and combine like terms.

$$3n^{2}+4n+1+\left(n^{2}+3n+2\right)\times 3=\left(n+2\right)\left(3n+1\right)\times 2$$

Use the distributive property to multiply n^{2}+3n+2 by 3.

$$3n^{2}+4n+1+3n^{2}+9n+6=\left(n+2\right)\left(3n+1\right)\times 2$$

Combine 3n^{2} and 3n^{2} to get 6n^{2}.

$$6n^{2}+4n+1+9n+6=\left(n+2\right)\left(3n+1\right)\times 2$$

Combine 4n and 9n to get 13n.

$$6n^{2}+13n+1+6=\left(n+2\right)\left(3n+1\right)\times 2$$

Add 1 and 6 to get 7.

$$6n^{2}+13n+7=\left(n+2\right)\left(3n+1\right)\times 2$$

Use the distributive property to multiply n+2 by 3n+1 and combine like terms.

$$6n^{2}+13n+7=\left(3n^{2}+7n+2\right)\times 2$$

Use the distributive property to multiply 3n^{2}+7n+2 by 2.

$$6n^{2}+13n+7=6n^{2}+14n+4$$

Subtract 6n^{2} from both sides.

$$6n^{2}+13n+7-6n^{2}=14n+4$$

Combine 6n^{2} and -6n^{2} to get 0.

$$13n+7=14n+4$$

Subtract 14n from both sides.

$$13n+7-14n=4$$

Combine 13n and -14n to get -n.

$$-n+7=4$$

Subtract 7 from both sides.

$$-n=4-7$$

Subtract 7 from 4 to get -3.

$$-n=-3$$

Multiply both sides by -1.

$$n=3$$